A minimal approach for the local statistical properties of a one-dimensional disordered wire

We consider a one-dimensional wire in gaussian random potential. By treating the spatial direction as imaginary time, we construct a `minimal' zero-dimensional quantum system such that the local statistical properties of the wire are given as products of statistically independent matrix elements of the evolution operator of the system. The space of states of this quantum system is found to be a particular non-unitary, infinite dimensional representation of the pseudo-unitary group, U(1,1). We show that our construction is minimal in a well defined sense, and compare it to the supersymmetry and Berezinskii techniques.